These are just some of the fragmented musings that I note down to help me structure my thinking, reasoning, and writing, and give me some vague sense of mental clarity. They are not meant to be complete, based on established knowledge, or coherent (to anyone but me). But if you find them somehow helpful…that’s great. Sharing these thoughts here with the hope of one day integrating them into a more complete manual for anyone who, like me, struggles to organize their brain.

1 Organization

1.1 Large-scale structure

  • domain (e.g., math) > branch (analysis) > field (e.g., calculus) > subfield (e.g., vector calculus)
  • field = area (e.g., probability) > area section (e.g., supervised learning) > chapter (e.g., conditional probability; linear regression) > chapter section (multiple linear regression) > chapter sub-section (numeric feature engineering)

2 Fundamental structure of ideas

  • ideas = unified collections of statements that are the building blocks of higher-order thought and ideas (abstractions like ‘topics’ and ‘concepts’)
  • ideas are, at their most basic (‘elemental’), abstract representations of things and the actions on them
  • more specifically, objects with certain properties and the operations — with certain properties — on them
  • in math terms: objects+properties/operations+properties; in linguistic terms: nouns+adjectives/verbs+adverbs

3 Basic building blocks of statements

(linguistic/mathematical/etc. expressions of ideas)

  1. first-order statements

    • = basic true/false statements (e.g., x is/is not y)

      • note: can think of a statement like this as expressing that an object possesses a property or ‘does something’, which is roughly equivalent to an equivalence between object and property or object and operation), sometimes with categorical qualifiers (e.g., all, some, etc.)

      • note also: can think of these as definitions

    • more complex definitions (e.g., of more complex objects/operations) may combine multiple of these using logical connectives (see (3))

    • we can call these ‘compound statements’ of first-order statements

  2. equivalences/implications

    • = combined first-order statements that are either equivalent to or follow from one another

      • note: can think of these as ‘theorems’ — where the equivalence (equality/inequality)/implication requires proof

      • note also: even first-order statements sometimes require proof (e.g., that an object/operation, or a particular subset or combination of objects/operations, has a particular property) = constructive proof

  3. higher-order statements

    • = combinations of (1) and (2) using logical connectives (and, or — more accurately, conjunctions, since ‘logical connectives’ includes ‘not’ [‘truth qualifier’])

      • note: first-order statements can also contain lists — e.g., of objects, attributes, or details

      • just be careful to not list a series of distinct ideas that act more as clauses than list items and end up with an extremely cluttered sentence

All of the above can be viewed as elaborations on basic object (modifier)-verb (modifier) structure fundamental to most (if not all) modern human language using the explanatory and reasoning tools of grammar and logic.

3.1 Argumentative structure

(applies best to argumentative pieces/mathematics/other complex ideas)

3.1.1 General principle

  • generally working from knowns -> unknowns through a series of logical connectives (equalities/implications)

3.1.2 Argumentative

(i.e., logical) structure (for expressing complex higher-order ideas)

  • topic sentence = claim (unknown); concluding sentence = conclusion = restatement of claim (after support given = known)
  • points = knowns (if self-evident or assumed; if implied by or equivalent to previous point)/unknowns otherwise (claim)
  • subpoints = knowns (if self-evident or assumed; if implied by or equivalent to previous point)

3.1.3 Proofs/derivations

(mathematical explanations)

  • note the relationship between argumentative structure and proof structure
  • in mathematics we prove statements (notes on this in ‘basic building blocks’)

3.2 General principles of composition

  • topics/points/subpoints can be arranged

    • in sequence
    • in parallel

3.3 Hierarchies and symmetries

3.3.1 Hierarchies

  • topic sentences encompass all points beneath
  • points encompass all subpoints beneath

3.3.2 Symmetries

  • symmetry = restatement of an idea

    • e.g., topic sentences are symmetrical to concluding sentences

  • points of a topic paragraph are symmetrical to the topic sentences of all support paragraphs